I have sequence of random variables defined by the following recursion:

$$X_{n+1} = X_n+\begin{cases} \alpha(S_n - X_n), \text{ if } S_n > X_n \\ \beta(S_n - X_n), \text{ if } S_n < X_n, \end{cases}$$ where $0<\beta < \alpha <1$ are constants, $(S_n)$ are i.i.d with known distributions. Also, $S_n$ independent of $\sigma(X_1, X_2,\dots, X_n)$ and $X_ 0 = 0.$

Initially, I asked about the convergence/limiting distribution of $X_n,$ but after doing some research, I realize that it is generally considered a very difficult problem - to obtain explicit distribution/asymptotics.

Therefore, I want to ask following questions with increasing orders of difficulties (according to my very limited probability theory knowledge.)

1) Can we at least prove that it has a limiting distribution? It looks like one can formulate this as a general state space Markov Chain but there do not seem to be an abundance of sources on this topic. Probability by Durrett has a brief chapter on it and he mentions that discrete Orstein- Uhlehnbeck process: $$V_{n+1} = \theta V_n+\xi_n$$ is an example of a discrete time, general state space Markov Chain. However, most of the resources I could find on the internet refers to the continuous one and as such my hope of modifying proofs for OU did not pan out.

2) If there is a limiting distribution, what kind of qualitative results can I hope to achieve? For example, one has the following for the expected value: $$\mathbb{E}[X_{n+1}] = \mathbb{E}[X_n](1 - \beta ) + \beta\mu + ( \alpha - \beta)\mathbb{E}[\delta_n\mathbb{1}_{\delta_n >0}],$$ where $\delta_n = S_n - X_n,$ and $\mu = \mathbb{E}[S_n].$ But then, the issue I am having is manipulating: $$P(S_n - X_n > t|S_n > X_n)$$, which will come from the last term.

I will greatly appreciate if anyone has some ideas or point me to a helpful source.

**Simulation:** I attach some simulations that *seem* to suggest that there is a bounded, limiting distribution.